Linear robustness analysis of non-linear biomolecular networks with - - PowerPoint PPT Presentation

Linear robustness analysis of non-linear biomolecular networks with kinetic perturbations

Steffen Waldherr and Frank Allgöwer

Institute for Systems Theory and Automatic Control Universität Stuttgart

MSC 2011, Workshop on Robustness in Biological Systems

Robustness analysis

Informal definition of robustness

Robustness is a property that allows a system to maintain its function in the presence of internal and external perturbations.

• H. Kitano, 2004

System / Model: ˙ x = Sv(x)

x ∈ Rn: amounts of signalling molecules v(x) ∈ Rm: reaction rates S ∈ Rn×m: stoichiometric matrix

Perturbations: kinetic perturbations (changes in reaction kinetics v) Function: Qualitative dynamical behaviour (stability, oscillations, ...)

Robustness analysis with kinetic perturbations, S. Waldherr 1 / 26

Formal robustness measures

robust perturbations

Smallest non-robust perturbation in ∞- norm loss of function

nominal point

Non-robust perturbation = any deviation from the nominal point for which the system looses functionality Robustness measure = smallest norm of non-robust perturbations

Robustness analysis with kinetic perturbations, S. Waldherr 2 / 26

Why robustness analysis?

Robustness analysis for model evaluation

Biological systems are robust ↔ models should be robust! Find parts of the system which need to be modelled carefully Evaluate effects of uncertainty on model predictions

Robustness analysis for system understanding

Understand mechanisms which confer robustness to biological systems Detect fragile points, e.g. for medical intervention Guide system design in bioengineering and synthetic biology

Robustness analysis with kinetic perturbations, S. Waldherr 3 / 26

Outline

1

Theory of kinetic perturbations

2

Robustness analysis with kinetic perturbations

3

Adaptation with kinetic perturbations

Robustness analysis with kinetic perturbations, S. Waldherr 4 / 26

Outline

1

Theory of kinetic perturbations Definition of kinetic perturbations Mapping kinetic perturbations to parameter variations

2

Robustness analysis with kinetic perturbations

3

Adaptation with kinetic perturbations

Robustness analysis with kinetic perturbations, S. Waldherr 5 / 26

Kinetic perturbations – the basic idea

Kinetic perturbations = reaction rate perturbations

nominal rates v(x) to perturbed rates ˜ v(x) ⇒ nominal model ˙ x = Sv(x) to perturbed model ˙ x = S˜ v(x) Consider nominal steady state x0: Sv(x0) = 0 Kinetic perturbation, if ˜ v(x0) = v(x0) Reaction rates in steady state unperturbed, but reaction rate slopes may vary:

x v(x)

1 2 1

v(x) = 0.7x ˜ v(x), increased slope ˜ v(x), reduced slope

x0 = 1

Robustness analysis with kinetic perturbations, S. Waldherr 6 / 26

Formal definition of kinetic perturbations

Definition

The network ˙ x = S˜ v(x) is said to be a kinetic perturbation of the nominal network ˙ x = Sv(x) at x0 ∈ Rn with Sv(x0) = 0, if ˜ v(x0) = v(x0) =: v0. Consider the reaction rate Jacobian V = ∂v

∂x

For the perturbed network, we have ˜ V(x0) = ∂˜ v ∂x (x0) = V(x0) + ¯ ∆ Local effects of the kinetic perturbation are determined by ¯ ∆

Scaled perturbation ∆iℓ =

x0,ℓ v0,i ¯

∆iℓ ∆ is a measure for the perturbation size Steady state concentration x0 and flux v0 are unperturbed Steady state reaction slopes V(x0) are perturbed by ¯ ∆

Robustness analysis with kinetic perturbations, S. Waldherr 7 / 26

Mapping kinetic perturbations to parameter changes

Considering generalised mass action (GMA) networks GMA reaction rate: vi(x) = ki n

ℓ=1 xαiℓ ℓ

αiℓ > 0: xℓ is a substrate for / activates vi αiℓ < 0: xℓ inhibits vi

Non-integer kinetic orders αiℓ

Effect of constrained diffusion (fractal reaction kinetics) Result of model simplifications (as in S-systems, Savageau et al.)

Perturbed reaction rate ˜ vi(x) = ˜ ki n

ℓ=1 x ˜ αiℓ ℓ

Kinetic perturbation

∂˜ vi ∂xℓ = ∂vi ∂xℓ + v0,i x0,ℓ ∆iℓ

⇔ Parameter change ˜ αiℓ = αiℓ + ∆iℓ ˜ ki = ki

n

• ℓ=1

x−∆iℓ

0,ℓ

Robustness analysis with kinetic perturbations, S. Waldherr 8 / 26

Biochemical interpretation of kinetic perturbations

Are kinetic perturbations biochemically plausible? Yes, because kinetic perturbations map to parameter variations in reasonable model classes for intracellular networks such as GMA networks enzymatic networks (Michaelis-Menten or Hill reaction kinetics) genetic networks (sigmoidal activation & inhibition functions)

Biochemical interpretation

Positive ∆iℓ: more cooperativity in reaction i with respect to species ℓ Negative ∆iℓ: increase saturation in reaction i with respect to species ℓ

Robustness analysis with kinetic perturbations, S. Waldherr 9 / 26

Outline

1

Theory of kinetic perturbations

2

Robustness analysis with kinetic perturbations Kinetic perturbations affect dynamical properties Robustness analysis via the structured singular value Example: Oscillations in the MAPK cascade

3

Adaptation with kinetic perturbations

Robustness analysis with kinetic perturbations, S. Waldherr 10 / 26

Inducing bifurcations by kinetic perturbations

Example “network”

X v2 v1 v1 =

kx2 1+Mx2

˙ x = v1 − v2 v2 = x

x v(x)

1 2 1

v2 v1

x0 = 1

∂˜ v1 ∂x (1) = ∂v1 ∂x (1) + ¯

∆ ¯ ∆ = 0: x0 stable ¯ ∆ = ¯ ∆∗: transcritical bifurcation at x0 ¯ ∆ > ¯ ∆∗: x0 unstable Loss of bistability by a kinetic perturbation

Robustness analysis with kinetic perturbations, S. Waldherr 11 / 26

Inducing bifurcations by kinetic perturbations

Example “network”

X v2 v1 v1 =

kx2 1+Mx2

˙ x = v1 − v2 v2 = x

x v(x)

1 2 1

v2 v1 ˜ v1

x0 = 1

∂˜ v1 ∂x (1) = ∂v1 ∂x (1) + ¯

∆ ¯ ∆ = 0: x0 stable ¯ ∆ = ¯ ∆∗: transcritical bifurcation at x0 ¯ ∆ > ¯ ∆∗: x0 unstable Loss of bistability by a kinetic perturbation

Robustness analysis with kinetic perturbations, S. Waldherr 11 / 26

Inducing bifurcations by kinetic perturbations

Example “network”

X v2 v1 v1 =

kx2 1+Mx2

˙ x = v1 − v2 v2 = x

x v(x)

1 2 1

v2 v1 ˜ v1

x0 = 1

∂˜ v1 ∂x (1) = ∂v1 ∂x (1) + ¯

∆ ¯ ∆ = 0: x0 stable ¯ ∆ = ¯ ∆∗: transcritical bifurcation at x0 ¯ ∆ > ¯ ∆∗: x0 unstable Loss of bistability by a kinetic perturbation

Robustness analysis with kinetic perturbations, S. Waldherr 11 / 26

Robustness with respect to kinetic perturbations

Perturbed Jacobian

˜ A(∆) = A + S(diag v0)∆(diag x0)−1

Robustness radius for kinetic perturbations

Robustness measure = size of smallest perturbation such that ˜ A(∆) has an eigenvalue on the imaginary axis ψ = inf

• ∆ | det(jωI − ˜

A(∆)) = 0

• Can be formulated as structured singular value (µ) problem with

G(jω) = (diag x0)−1(jωI − A)−1S(diag v0) Result: ψ =

• sup

ω

µ∆G(jω) −1 G(jω) ∆ Robustness problem can be solved with standard µ analysis tools.

Robustness analysis with kinetic perturbations, S. Waldherr 12 / 26

Robustness radius for scalar perturbations

Perturbation of a single interaction: Species ℓ on reaction i ¯ ∆ =

        . . . 1 . . .        

¯ ∆iℓ(0 . . . 1 . . . 0) = ei ¯ ∆iℓeT

ℓ

Perturbed Jacobian ˜ A(∆iℓ) = A + Seiv0,i∆iℓx−1

0,ℓ eT ℓ

Robustness radius

ψ =

• |∆iℓ| | det(jωI − ˜

A(∆iℓ)) = 0

• =
• sup

ω∈R(Gℓi)

|Gℓi(jω)| −1

Re Im

Gℓi(jω) 1 ψ

µ–problem with Gℓi(jω) = eT

ℓx−1 0,ℓ (jωI − A)−1Sv0,iei

Explicit formula for the robustness radius Allows to detect fragile interactions in the network

Robustness analysis with kinetic perturbations, S. Waldherr 13 / 26

Robustness analysis in the vector case

Perturbation of several interactions either

from one species ℓ to several reactions Eiv or from several species Eℓx to one reaction i.

¯ ∆ = Ei ¯ ∆iℓeT

ℓ

Perturbed Jacobian ˜ A(∆iℓ) = A + SEi diag(Eiv0)∆iℓx−1

0,ℓ eT ℓ

Robustness analysis in the vector case

Results depend on the norm that is chosen:

1-norm or ∞-norm: robustness radius from a linear program 2-norm: explicit formula for robustness radius

Also requires to compute a supremum over ω. Non-robust perturbation is the solution of an affine equation M ¯ ∆ = b.

Robustness analysis with kinetic perturbations, S. Waldherr 14 / 26

The MAPK cascade

E2 INPUT (E1) MAPKKK MAPKKK* MAPKK P’ase MAPKK−P MAPKK MAPKK−PP MAPK MAPK−P MAPK−PP MAPK P’ase OUTPUT

Huang & Ferrell 1996 Step response for MAPKpp

time [min] MAPKpp [µM]

200 400 600 0.1 0.2

Medium scale network with 22 species and 30 (irreversible) reactions Where to perturb this network in order to get oscillations?

Robustness analysis with kinetic perturbations, S. Waldherr 15 / 26

Single interaction analysis of the MAPK cascade

Checking all 660 network interactions for a scalar kinetic perturbation to induce oscillations Found 8 fragile interactions (|∆∗| < 1), e.g.:

E2 INPUT (E1) MAPKKK MAPKKK* MAPKK P’ase MAPKK−P MAPKK MAPKK−PP MAPK MAPK−P MAPK−PP MAPK P’ase OUTPUT

Negative feedback from MAPKpp to MAPKK phosphorylation Positive feedback from MAPKpp to catalysis of MAPK / MAPKKpp complex Saturated catalysis of MAPKpp / Phosphatase complex

Robustness analysis with kinetic perturbations, S. Waldherr 16 / 26

Single interaction analysis of the MAPK cascade

Checking all 660 network interactions for a scalar kinetic perturbation to induce oscillations Found 8 fragile interactions (|∆∗| < 1), e.g.:

E2 INPUT (E1) MAPKKK MAPKKK* MAPKK P’ase MAPKK−P MAPKK MAPKK−PP MAPK MAPK−P MAPK−PP MAPK P’ase OUTPUT

Negative feedback from MAPKpp to MAPKK phosphorylation Positive feedback from MAPKpp to catalysis of MAPK / MAPKKpp complex Saturated catalysis of MAPKpp / Phosphatase complex

Robustness analysis with kinetic perturbations, S. Waldherr 16 / 26

Single interaction analysis of the MAPK cascade

Checking all 660 network interactions for a scalar kinetic perturbation to induce oscillations Found 8 fragile interactions (|∆∗| < 1), e.g.:

E2 INPUT (E1) MAPKKK MAPKKK* MAPKK P’ase MAPKK−P MAPKK MAPKK−PP MAPK MAPK−P MAPK−PP MAPK P’ase OUTPUT

Negative feedback from MAPKpp to MAPKK phosphorylation Positive feedback from MAPKpp to catalysis of MAPK / MAPKKpp complex Saturated catalysis of MAPKpp / Phosphatase complex

Robustness analysis with kinetic perturbations, S. Waldherr 16 / 26

Single interaction analysis of the MAPK cascade

Checking all 660 network interactions for a scalar kinetic perturbation to induce oscillations Found 8 fragile interactions (|∆∗| < 1), e.g.:

E2 INPUT (E1) MAPKKK MAPKKK* MAPKK P’ase MAPKK−P MAPKK MAPKK−PP MAPK MAPK−P MAPK−PP MAPK P’ase OUTPUT

Negative feedback from MAPKpp to MAPKK phosphorylation Positive feedback from MAPKpp to catalysis of MAPK / MAPKKpp complex Saturated catalysis of MAPKpp / Phosphatase complex

Robustness analysis with kinetic perturbations, S. Waldherr 16 / 26

Effect of a kinetic perturbation on the dynamics

MAPK K-P’ase MAPK K-P’ase

+

P P P

v30

Critical perturbation ∆∗ = −0.76

substrate v30

0.025 µM 4 µM

min

∆ = 0

x0

time [min] MAPKpp [µM]

200 400 600 0.4 0.8

Robustness analysis with kinetic perturbations, S. Waldherr 17 / 26

Effect of a kinetic perturbation on the dynamics

MAPK K-P’ase MAPK K-P’ase

+

P P P

v30

Critical perturbation ∆∗ = −0.76

substrate v30

0.025 µM 4 µM

min

∆ = 0 ∆ = −0.4

x0

time [min] MAPKpp [µM]

200 400 600 0.4 0.8

Robustness analysis with kinetic perturbations, S. Waldherr 17 / 26

Effect of a kinetic perturbation on the dynamics

MAPK K-P’ase MAPK K-P’ase

+

P P P

v30

Critical perturbation ∆∗ = −0.76

substrate v30

0.025 µM 4 µM

min

∆ = 0 ∆ = −0.4 ∆ = −0.8

x0

time [min] MAPKpp [µM]

200 400 600 0.4 0.8

Robustness analysis with kinetic perturbations, S. Waldherr 17 / 26

Outline

1

Theory of kinetic perturbations

2

Robustness analysis with kinetic perturbations

3

Adaptation with kinetic perturbations Properties of biomolecular adaptation Achieving adaptation with kinetic perturbations Example: Adaptation in the MAPK cascade

Robustness analysis with kinetic perturbations, S. Waldherr 18 / 26

Definition of adaptation

• 1. Adaptation

A dynamical system u → y is said to adapt, if persistent changes in u do lead to transient, but not persistent changes in y.

Stimulus Response Time

Local adaptation = consider only infinitesimally small changes in u

• 2. Local adaptation without transient response

The system u → y is said to perfectly adapt locally at (u0, y0), if lim

us→0

limt→∞(y(t, u0 + ush(t)) − y0) us = 0.

Robustness analysis with kinetic perturbations, S. Waldherr 19 / 26

Adaptation and robustness

Adaptation to a slowly changing perturbation = Robustness of steady state concentration to this perturbation

Perturbation Response Time

Slowly changing perturbations in biology

Protein amounts due to stochastic gene expression Environmental conditions such as light, temperature, or nutrient availability

Robustness analysis with kinetic perturbations, S. Waldherr 20 / 26

Conditions for adaptation

• 1. Linear approximation at steady state

˙ dx = Adx + Bdu dy = Cdx, with: A = S ∂v ∂x (x0, u0) B = S ∂v ∂u (x0, u0)

• 2. Tranformation to frequency domain

dy(s) = C(sI − A)−1Bdu(s) = G(s)du(s).

• 3. Necessary and sufficient condition for local adaptation

By the final value theorem of the Laplace transformation: local adaptation ⇔ G(0) = 0

A invertible

⇔ det A B C

• = 0

Last condition is affected by kinetic perturbations!

Robustness analysis with kinetic perturbations, S. Waldherr 21 / 26

Achieving adaptation with kinetic perturbations

Effect of kinetic perturbation on adaptation condition

Perturbation of Jacobian due to a kinetic perturbation ℓ → i: ˜ A = A + Seiv0,i∆iℓ(x0,ℓ)−1eT

ℓ

Adaptation as goal: det ˜ A B C

• = det
• A + Seiv0,i∆iℓ(x0,ℓ)−1eT

ℓ

B C

• = 0.

Compute a critical kinetic perturbation

∆∗

iℓ = (−eT j MoutM−1 0 Mineℓ)−1 x0,ℓ

v0,i with

M0 = A B C

• Min =

I

• S

Mout =

• I
• Robustness analysis with kinetic perturbations, S. Waldherr

22 / 26

Example: Adaptation in the MAPK cascade

Goal: achieve adaptation of nuclear MAPKpp

E1 KKK KKKp E2 KK KKp KKpp KKPase K Kp Kpp KPase Kpp

u y

cytosol nucleus

Huang-Ferrell MAPK cascade model as before Added an input by controlling E1 production

Result of analysis with kinetic perturbations

No adaptation with transient response possible through kinetic perturbations Network lacks a feedback or feedforward loop

Robustness analysis with kinetic perturbations, S. Waldherr 23 / 26

Adaptation in an extended MAPK cascade model

Model feedback extension

E1 KKK KKKp E2 KK KKp KKpp KKPase K Kp Kpp KPase Kpp RP-gene TC RP-mRNA RP IC

u y

cytosol nucleus

Added a protein genetically regulated by MAPK Feedback by formation

• f inhibitory complex

Candidate interactions for adaptation

Found five candidate interactions in newly added feedback loop for adaptation via kinetic perturbations.

Robustness analysis with kinetic perturbations, S. Waldherr 24 / 26

Response of original and perturbed networks

Adaptation achieved even for large stimulus changes. Non-linear dynamics depend on which interaction is targeted.

Robustness analysis with kinetic perturbations, S. Waldherr 25 / 26

Conclusions

Kinetic perturbations

Perturbation class suited for biochemical network analysis Kinetic perturbations = reaction rate slope changes at steady state Biochemically plausible for GMA / enzymatic / genetic networks

Robustness analysis with kinetic perturbations

Efficient robustness analysis via µ–analysis methods Results indicate fragile network interactions Non-robust perturbations are explicitly computed

Adaptation with kinetic perturbations

Find network modifications to achieve perfect local adaptation Steady state is kept unperturbed intrinsically Adaptation even to large stimulus changes in many cases

Robustness analysis with kinetic perturbations, S. Waldherr 26 / 26